As others have pointed out, there is no clear way to definitively answer this question because of the number of issues involved and no clear precedent.
For anyone curious, it may, however, be helpful to add information about which issues may or may not contribute to answering this question if it ever reaches the courts.
I will try to answer this without rehashing what's already been stipulated in the question itself or repeating (rather than referring to) the other answers which already tried to giver an overview.
As mentioned in this answer, Hammerhead Enterprises, Inc. v. Brezenoff was decided based on the facts that the government did not have any retaliatory authority and that the stores did not actually change their behavior in response to the purported government intimidation.
Which was distinct from Bantam v. Sullivan decision,
Such notices requested the distributor's "cooperation," and advised him that copies of the lists of "objectionable" publications were circulated to local police departments, and that it was the Commission's duty to recommend prosecution of purveyors of obscenity. Four out-of-state publishers of books widely distributed in the State sued in a Rhode Island court for injunctive relief and a declaratory judgment that the law and the practices thereunder were unconstitutional. The court found that the effect of the Commission's notices was to intimidate distributors and retailers and that they had resulted in the suppression of the sale of the books listed.
The parallel is fairly clear here. If Facebook is a distributor of information, then requests to a distributor to quash ill-advised speech amount to intimidation.
But what happens to the claims that Facebook actually solicits this information from the Government to better shape its own platform?
This is where a timeline might get tricky.
First, is it possible for political intimidation to become a deciding factor?
Well, Trump's 1st executive order on the "Muslim ban" was decided based on his campaign speeches as well as the text of the order itself.
So while, political speech is protected, its effect does not have to be ignored when deciding whether another party's rights have been diminished due to a different action (rather than the speech itself). The decision did not reach SCOTUS because a new executive order was issued, which made the issue irrelevant.
But lower courts did have a chance to state that prior political speeches could be used in interpreting an executive action.
There is little doubt that there is a political campaign to add regulation to social media sites by both political parties.
Even the CEOs of the social media sites have been subpoenaed to testify about their sites in front of Congress. Some of the questions posited appeared to be chastising rather than formulating genuine inquiries.
The timeline question, which may end up deciding this, would be when the reporting tool was created by Facebook. If the tool was created after numerous political threats, then the Government's use of this tool can be easily interpreted as Facebook acting, while under a threat of reprisal.
However, what if it can be shown that the tool existed, in its current form, even before any political threats were made?
Then such a claim, that this request for information (from the government by Facebook) amounted to intimidation, would seem unwarranted.
What if some political voices were heard trying to speak out against FB's political influence, but they were not loud or numerous when the tool was created?
Well, then the decision, on whether the Government did strong arm Facebook to enlist it in quashing ill-advised speech, would have to be resolved by a court.
Since Facebook would be viewed a distributor in such a case, a likely petitioner would be a professional entity which saw its professional speech suppressed. The defendant would not be Facebook itself, however, but a government official who used the Facebook's tool to request the suppression.